Convergence Rates for a Finite Volume Scheme of the Stochastic Heat Equation

Lemma A.1.

Let d ∈ { 2 , 3 } . Then, for any u ∈ H 2 ⁢ ( Λ ) satisfying the weak homogeneous Neumann boundary condition we have

where Δ denotes the Laplace operator on H 1 ⁢ ( Λ ) associated with the weak formulation of the homogeneous Neumann boundary condition. Especially, for any random variable u : Ω → H 2 ⁢ ( Λ ) we obtain

Proof.

Let u ∈ H 2 ⁢ ( Λ ) satisfying the weak homogeneous Neumann boundary conditions. We set

Then, according to [23, Theorem 3.2.1.3], u is the unique solution of - Δ ⁢ u + u = f satisfying the weak homogeneous Neumann boundary conditions. Now, we follow the ideas of the proof of [23, Theorem 3.2.1.3]. There exists a sequence ( Λ m ) m such that Λ m ⊂ ℝ d is bounded, open, convex, Λ ⊂ Λ m , dist ⁡ ( ∂ ⁡ Λ , ∂ ⁡ Λ m ) → 0 as m → ∞ and Λ m has a 𝒞 2 -boundary for any m ∈ ℕ . We set

Then, since f ~ ∈ L 2 ⁢ ( Λ m ) for any m ∈ ℕ , there exists a unique u m ∈ H 2 ⁢ ( Λ m ) satisfying the weak homogeneous Neumann boundary conditions with respect to Λ m such that - Δ ⁢ u m + u m = f ~ in Λ m . According to the proof of [23, Theorem 3.2.1.3] there exists C > 0 such that ∥ u m ∥ H 2 ⁢ ( Λ m ) ≤ C for any m ∈ ℕ and ( u m ) | Λ ⇀ u in H 2 ⁢ ( Λ ) for a subsequence. Now, [23, Theorem 3.1.2.3] yields

Hence, we obtain

and therefore we may conclude

Lemma A.2.

Let u ∈ H 2 ⁢ ( Λ ) , T an admissible mesh, K ∈ T and y ∈ K . Then, for any 1 ≤ q ≤ 2 , there exists a constant C = C ⁢ ( q , Λ ) > 0 such that

Especially, for any function v of the form v ⁢ ( x ) := ∑ K ∈ T u ⁢ ( y K ) ⁢ 1 K ⁢ ( x ) , where y K ∈ K and x ∈ Λ , we have

Proof.

Let x , y ∈ K . Then we have

The Jensen inequality yields

Now, a change of variables z := ( 1 - s ) ⁢ y + s ⁢ x yields

for C = C ( q , Λ ) = 2 q - 1 ( 1 + diam ( Λ ) q ) . ∎